Timber Design for Bridges

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BASIC TIMBER DESIGN CONCEPTS FOR BRIDGES

5.1 INTRODUCTION

For thousands of years, timber bridges and other timber structures werebuilt primarily by trial and error and rule of thumb. Designs were based onpast experience, and little concern was given to efficient material usage oreconomy. As the complexity of structures increased, more attention wasfocused on the importance of accurate engineering methods. Research wasundertaken to develop design criteria for wood with the same level ofaccuracy and reliability available for other engineering materials. As aresult, developments in timber design have advanced substantially in thiscentury. Although wood is orthotropic and differs in many respects fromother materials, wood structures are designed using many of the sameequations of mechanics developed for isotropic materials. Variations inmaterial properties from growth characteristics, manufacturing, and useconditions are compensated for by material grading and stress adjustmentsapplied in the design process. Timber design may seem confusing at first,but with experience it is no more difficult than design with other materials.

This chapter provides an overview of basic design concepts for sawnlumber and glulam used in bridge design. It includes specification requirements and methods for designing beams, tension members, columns,combined axial and bending members, and connections. Applications ofthese concepts to design situations are given in examples for each memberand connection type. More detailed design related to specific bridge typesis covered in Chapters 7, 8, and 9.

The discussions and examples in this chapter are based on a number ofreferenced specifications that were current at the time of publication. Thereader is cautioned to verify these requirements against the most recentedition of the specifications before designing a bridge. In no case shouldthe information presented in this chapter be considered a substitute for themost current design specifications.

5.2 DESIGN SPECIFICATIONS AND STANDARDS

The primary specifications for bridge design in the United States are theStandard Specifications for Highway Bridges, adopted and published bythe American Association of State Highway and Transportation Officials(AASHTO).

lThese specifications are published intermittently and are

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revised annually through the issuance of interim specifications. Theyaddress all areas of bridge design, including geometry, loading, and designrequirements for materials. AASHTO specifications are used extensivelyas the standard for bridge design and are the primary reference for thetimber design requirements, procedures, and recommendations addressedin this manual.

The majority of the timber design requirements in AASHTO are based ontheNational Design Specification for Wood Construction (NDS).

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TheNDS is the most widely recognized general specification for timber designand is published periodically by the National Forest Products Association.The specification includes design requirements and tabulated designvalues for sawn lumber, glulam, and timber piles. Although the NDS doesnot specifically address detailed bridge design, it does serve as the basisfor the timber design concepts and requirements used for bridges. Notationof the NDS as the source of design requirements in this chapter reflectsreferences in AASHTO that specify the NDS as the most current source oftimber design information for bridges (AASHTO 13.1.1).

In addition to the NDS, AASHTO periodically references the specifications, standards, and technical publications of the American Institute ofTimber Construction (AITC). AITC is the national technical trade association of the glulam industry and is responsible for numerous specificationsand technical publications addressing fabrication, design, and constructionof glulam. AITC also publishes AITC 117-Design Standard Specifications for Structural Glued Laminated Timber of Softwood Species (AITC117-Design), which is the source of tabulated values for glulam.

4

Timber design requirements for bridges may differ from those commonly

used for buildings and other structures. Although the requirements inAASHTO are based on the NDS and other referenced specifications andstandards, modifications have been incorporated in AASHTO to addressspecific bridge requirements. The designer should become familiar withthe content and requirements of current AASHTO, NDS, and AITCspecifications. Copies of these specifications and other noted referencesare available from the parent organizations at the addresses listed inTable 16-10.

5.3 DESIGN METHODS AND VALUES

Timber bridges are designed according to the principles of engineeringmechanics and strength of materials, assuming the same basic linearelastic theory applied to other materials. The method used for design is theallowable stress design method, which is similar to service load design forstructural steel. In this method, stresses produced by applied loads must be

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TABULATED DESIGNTABULATED DESIGNVALUESVALUES

Table 5-1.- Stress symbols for timber components.

values about thex-x ory-y axis of the member (thex-x axis for glulam isalways parallel to the wide face of the laminations). For example, Fbx isthe tabulated bending stress about the x-x axis. In the absence of such asubscript, it is assumed that stresses act about thex-x axis.

Tabulated design values for sawn lumber and glulam are based on testingand grading processes discussed in Chapter 3. These values represent themaximum permissible values for specific conditions of use and normallyrequire adjustments for actual design conditions. In this sense, tabulatedvalues should be viewed only as the basis or starting point for determiningthe allowable values to be used for design. An abbreviated summary oftabulated values for sawn lumber and glulam is published in AASHTO;however, these values do not include all species and grades and may notbe current. For this reason, AASHTO requires that tabulated values comply with those specified in the most current edition of the NDS or AITCspecifications (AASHTO 13.1.1 and 13.2.2). The source of tabulatedvalues for sawn lumber isDesign Values for Wood Construction, which isan integral part of the NDS, but is published as a separate volume. Tabulated values for glulam are given in AITC 117-Design. These NDS andAITC specifications represent the most comprehensive and current sourceof design information and include tabulated values for the followingproperties:

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End grain in bearing (Fg) Modulus of elasticity (E)

Tabulated Values for Sawn LumberTabulated values for visually graded and machine stress rated (MSR) sawnlumber are published in the NDS based on the grading rules established byseven grading agencies. Separate tables are included for visually graded

sawn lumber, MSR lumber, and end grain in bearing. The values are validfor sawn lumber used in dry applications under normal loading conditions(both of these conditions are discussed later for modification factors). Inaddition, each table contains an extensive set of footnotes for adjustingvalues to specific use conditions.

Visually Graded Sawn LumberDesign values for visually graded sawn lumber are specified in Table 4Aof the NDS. A portion of this table is shown in Table 5-2. The table gives

classification, and commercial grade of the lumber. When using the table,tabulated values for Fb, Ft, Fv, Fc, and Ebased on the species, size

the following considerations will help interpret tabulated values:1. Wood species may be specified as an individual species or a

species combination. When species combinations are used, theindividual species of the combination are listed in the Table 4Atable of contents.

2. The grading rules agencies for each species are noted in the farright column of the tables. When grading rules for the samespecies differ among agencies, tabulated values are givenseparately for each grading agency.

3. Tabulated values for each species are based on the grade and sizeclassification. Although commercial grade designations may bethe same, tabulated values can vary among size classifications.For example, the tabulated values for grade No. 1 in the Beamsand Stringers (B&S) size classification are not necessarily thesame as those for No. 1 in the Posts and Timbers (P&T) sizeclassification.

4. For all dimension lumber that is 2 to 4 inches thick, grading rulesand commercial-grade nomenclature are standardized. When sawnlumber is thicker than 4 inches, grades are not standardized, andtabulated values for the same species, size, and grade of member

may vary among grading agencies. In situations where conflictingtabulated values are given for different agencies, the designermust either specify the grading rules agency or use the lowertabulated values.

5. The availability of sawn lumber in the species, grade, and sizeclassifications in Table 4A of the NDS may be geographicallylimited. The designer should verify availability before specifyinga particular species, size, or grade.

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Tabulated Values for Glued-Laminated Timber (Glulam)Tabulated values for glulam are specified in AITC 117-Design. Separatetables are included for bending combinations, axial combinations, and endgrain in bearing. Values are given for western species and Southern Pinemade with either visually graded or E-rated lumber based on dry-useconditions (moisture content of 16 percent or less) and normal duration ofload. Tabulated values for a specific combination symbol of glulam arestandardized and are not subject to variations in grading rules or fabrication

processes.

Bending CombinationsFor bending combinations, tabulated values are given in Table 1 ofAITC117-Design. The combination symbols in this table are for membersconsisting of four or more laminations, stressed primarily in bending withloads applied perpendicular to the wide faces of the laminations (x-x axis).The table also includes tabulated values for axial loading and bending withloads applied parallel to the wide faces of the laminations (y-y axis);however, the axial combinations are usually better suited for these loadingconditions. A limited number of combination symbols, taken from Table 1from AITC 117-Design,are shown in Table 5-4. The first two columns ofthe table give the combination symbol and species of the member. Theremainder of the table is divided into three parts based on the type anddirection of applied stress. Columns 3 to 8 contain stresses for membersloaded in bending about the x-x axis (the most common case). For thiscondition, stresses for Fb and are specified separately for the tensionand compression zones of the member. These stresses may be the same forboth zones (balanced combination) or may differ significantly. Columns 9to 13 are for members loaded in bending about the y-y axis where stressesin the tension and compression zones are equal. Columns 14 to 16 are formembers loaded axially or with a combination of axial and bending loads.The intended use and limitations for groups of combinations are also notedin the table.

Axial CombinationsTabulated values for axial combinations are specified in Table 2 ofAITC117-Design. The combinations in this table are intended primarily formembers loaded axially or in bending with loads applied parallel to thewide faces of the laminations (y-y axis). The table also includes tabulatedvalues for loading perpendicular to the wide faces of the laminations (x-xaxis), but bending combinations are usually better suited for this condition.A limited number of combination symbols, taken from Table 2 fromAITC117-Design, are shown in Table 5-5. The table is organized in three sections based on the type and direction of applied stresses, as in Table 5-4.Tabulated values depend on the number of laminations and are given formembers consisting of 2, 3, and 4 or more laminations. For all axial combinations, strength properties are balanced about the neutral axis, and tabulated stresses for Fb and are equal in the tension and compression zones.

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Table 5-4.- Typical tabulated values for glulam bending combinations.

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ADJUSTMENTS TOTABULATED DESIGNVALUES

End Grain in BearingTabulated stress for end grain in bearing parallel to grain (Fg) is given inAnnex A ofAITC 117-Design. Annex A consists of Tables A-1 and A-2,which specify Fg for bending combinations and axial combinations, respectively. In both tables, Fg is specified by a combination symbol wheremember bearing is on the full cross section and where bearing is on apartial cross section.

Tabulated values for sawn lumber and for glulam are based on the standardconditions noted in the applicable design tables. When actual use conditions vary from these standard conditions, tabulated values must be ad

justed to compensate for (1) differences between the assumptions used toestablish tabulated values and actual use conditions, (2) variations in woodbehavior related to the type of stress or member orientation, and (3) differences between the physical or mechanical behavior of wood and that of anideal material assumed in most equations of engineering mechanics.

Requirements for adjusting tabulated values are given in the text of the

design specifications (AASHTO, NDS, and AITC 117-Design) and asfootnotes to tabulated values. The type and magnitude of the adjustments,as well as the manner in which they are applied, vary with the type ofmaterial, strength property, and design application. Most adjustments areapplied as modification factors that are multiplied by the tabulated values.These modification factors are designated by the letter C, followed by asubscript to denote the type of modification. They include the following:

CMmoisture content factor CL lateral stability of beams factor

CD duration of load factor CP lateral stability of columns factor

Cttemperature factor CR fire-retardant treatment factor

Cfform factor CCcurvature factor

CFsize factor CIinteraction stress factor

Modification factors are applied to tabulated values only, not to appliedstresses or loads. In most cases they are cumulative; however, in somecases the more restrictive value of two factors is used. A summary of theapplicability of modification factors to various wood properties is given inTable 5-6. The factors CCand CIapply to curved and taper-cut glulambeams, respectively, and are not discussed in this chapter. Refer to theAITC Timber Construction Manual for additional information on thesefactors.

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Moisture Content Factor CM)The strength and stiffness of wood decrease as moisture content increases.To compensate for this effect, tabulated values are adjusted by CM. Thisfactor, which is also referred to as a wet-use factor or condition-of-use

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Table 5-6.- Applicability of modification factors for strength properties and modulus of elasticity.

factor, is applicable to all tabulated values for strength and modulus ofelasticity. It adjusts values for changes in strength and stiffness and compensates for variations in cross section caused by shrinkage.

C

Application ofCMdiffers for sawn lumber and glulam. For sawn lumber,tabulated values are based on the moisture content specified for eachspecies in the NDS tables. With the exception of Southern Pine and Virginia Pine-Pond Pine, adjustment by CMis applied when the moisturecontent of the member in service is expected to exceed 19 percent. ForSouthern Pine and Virginia Pine-Pond Pine, the CMadjustment is notrequired because tabulated values are given in the design tables for three

in-service moisture contents. These tabulated values already include theMadjustment, and no further adjustment for moisture is required. Values

ofCMfor all other lumber species are given in the footnotes to the designtables and depend on the member size and specific strength property(Table 5-7).

For glulam, all tabulated values in AITC 117-Design are based on amoisture content in service of 16 percent or less. When the moisturecontent in service is expected to be 16 percent or higher, tabulated valuesmust be multiplied by the wet-use factors given in the design tables.Factor CMfor glulam depends on the strength property only and is inde

pendent of species, combination symbol, and member size. Values of CMfor glulam are given in Table 5-7.

In most applications, bridge members are exposed to the weather andshould be adjusted by CMfor wet-use conditions. In cases where beams areprotected by a waterproof deck, design for dry conditions may be appropriate, as discussed in Chapter 7.

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Table 5-7. - Values of the moisture content factorCMfor sawn lumber and glulam.

Duration of load Factor (CD)Wood is capable of withstanding much greater loads for short durationsthan for long periods. This is particularly significant in bridge designwhere short-term increased loads from vehicle overloads, wind, earthquake, or railing impact must be considered. The tabulated values forsawn lumber and glulam are based on an assumed normal duration ofload. In this case, a normal duration of load is based on the expectationthat members will be stressed to the maximum stress level (either continuously or cumulatively) for a period of approximately 10 years, stressed to90 percent of the maximum design level continuously for the remainder of

the life of the structure, or both. This maximum stress is assumed to occurduring the life of the member as a result of either continuous loading or aseries of shorter duration loads that total 10 years. When the maximumdesign loads act for durations that are shorter or longer than these assumeddurations, tabulated stresses are adjusted by CD, (Table 5-8). Factor CDapplies to tabulated strength properties but does not apply to compressionperpendicular to grain or modulus of elasticity (E). In most bridge

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Table 5-8. Modification factors for duration of load.

Duration of loadLoad duration factor CD

2 months (as for snow and ice) 1.157 days (as for snow and ice) 1.25

Wind or earthquake 1.335 minutes (rail loads only) 1.65

a

a

The duration of load factor for impact does not apply to members pressure-impregnated with preservative salts to the heavy retentions required for marine exposure, or sawn lumber treated with fire-retardant chemicals.

From AASHTO Section 13.2.5.1:188 1983. Used by permission.

applications, the permanent load of the structure is small in relation tovehicle loads, and a decrease in tabulated stresses for permanent loading isnot necessary

The stresses produced in bridge members are commonly the result of acombination of loads rather than a single load (Chapter 6). For a combination of loads of different durations, CD for the entire group is the singlevalue associated with the shortest load duration. When applying CD, thedesigner must recognize that for a given combination of loads, the mostrestrictive allowable stress may result from a partial combination involving loads of longer duration. The individual loads in a load combinationmust be evaluated in various combinations, with the value of CD depending on the load of shortest duration for that combination. This is accomplished by progressively eliminating the load of shortest duration from the

group and applying CD for the load of next-shortest duration. In otherwords, the resulting size or capacity of a member required for a loadcombination must not be less than that required for a partial combinationof the longer-duration loads. Application ofCD is discussed in more detailin Appendix B of the NDS and in Chapter 6. Duration of load is generallynot applicable in bridge design, except for the design of railing systems.

Temperature Factor (Ct)The strength and stiffness of wood increases as it cools and decreases as itwarms. These changes in strength because of temperature occur immediately and depend on the magnitude of the temperature change and the

moisture content of the wood. For temperatures up to approximately150OF, the immediate effects of strength loss are reversible, and the mem

ber will essentially recover its initial strength levels as the temperature islowered. Prolonged exposure to temperatures higher than 150

OF may

cause a permanent and irreversible loss in member strength.

Tabulated design values for sawn lumber and glulam assume that members will be used in normal temperature applications and may occasionally

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be heated to temperatures up to 150OF. This applies to most bridge design

situations. In cases where a member may be periodically exposed toelevated temperatures, humidity is generally low, and the increase inmember strength that results from reduced moisture tends to offset thereduction in strength that results from temporary temperature increases.The design specifications do not require a mandatory adjustment to tabulated values for temperature effects, and as a general rule, none are war

ranted. In cases where members will be exposed to prolonged temperatures in excess of 150

OF, or will be used at very low temperatures for the

entire design life, the modification factor, Ct, given in Table 5-9, may beapplied at the discretion of the designer.

Table 5-9. - Temperature factorCgiven as a percentage increase ortdecrease in design values for each 1

OF decrease or increase

in temperature.

C

Fire-Retardant Treatment Factor (CR)Fire-retardant treatments are seldom used on bridge members and areunnecessary in most applications. For those situations where fire-retardantchemicals are considered necessary, tabulated values must be adjusted bythe fire-retardant treatment factor CR. The value for this factor depends onspecific strength properties and is different for sawn lumber and glulam.

R is given for sawn lumber in Table 2A of the NDS (Table 5-10). Thebasis for these values and treatment qualifications are outlined in Appen

dix Q of the NDS. CR for glulam depends on the species and treatmentcombinations involved. The effects on strength properties must be determined for each treatment. However, indications are that 10 to 25 percentreductions in bending strength are applicable.

4,6The treatment manufac

turer should be contacted for more specific CR values for glulam based onthe specific material and design application.

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Table 5-10.- Fire-retardant treatment factor for structural lumber.

Property CRExtreme fiber in bending 0.85Tension parallel to grain 0.80Horizontal shear 0.90

Compression perpendicular to grain 0.90Compression parallel to grain 0.90Modulus of elasticity 0.90Fastener design loads 0.90

From the NDS;2688 1986. Used by permission.

Size Factor (CF)Tabulated bending stresses are based on a square or rectangular member

12 inches deep in the direction of applied loads. For member depthsgreater than 12 inches, Fb must be adjusted by CF, as computed by

(5-1)

where dis the member depth in inches.

For sawn lumber, CFdoes not apply to MSR lumber or to visually gradedlumber 2 to 4 inches thick used edgewise. For glulam, the CFvalue computed by the above equation is based on a uniformly distributed load on asimply supported beam with a span to depth ratioL/d= 21. In most bridge

applications, these assumptions result in reasonable accuracy as variationsin loading andL/dresult in relatively small deviations in the size factor. Incases where greater accuracy is warranted, CFmay be adjusted for other

L/dratios or loading conditions by the percentages in Table 5-11.

The effect of the size factor for both sawn lumber and glulam is to reducethe tabulated bending stress for members more than 12 inches deep. Formembers less than 12 inches deep, footnotes to design tables allow anincrease in bending stress for sawn lumber members 2 to 4 inches thickused flatwise,

24and glulam members loaded parallel to the wide faces of

the laminations.4CFis generally cumulative with other modification

factors, but is normally not cumulative with the lateral stability of beamsfactor, CL (see Sections 5.4 and 5.7).

Equation 5-1, used for computing size factor, is being reevaluated forglulam, and alternate forms of the equation are being considered byseveral industry-related technical committees. Thus, the designer shouldbe aware of the potential for future revisions and refer to the latest editionsof the NDS and AITC 117-Design for current requirements.

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Lateral Stabil ity of Beams Factor (CL)The lateral stability of beams factor, CL, is applied to some bending members where the compressive stress in bending must be limited to preventlateral buckling. Additional details on the use ofCL are discussed inSection 5.4.

Form Factor (C)fTabulated bending stresses are based on members with a square or rectangular cross section loaded normal to one or more faces. For other membershapes, specifically round or diamond sections, stresses must be modified

by the form factor, Cf. Cfdoes not apply to rectangular or square membersand is not commonly used in bridge applications. Refer to the NDS foradditional information on the use of Cf.

Lateral Stability of Columns Factor (CP)The lateral stability of columns factor, CP, is applied to some compressionmembers where the compressive stress must be limited to prevent lateralbuckling. Additional details on the use ofCP are discussed in Section 5.6.

A beam is a structural component with loads applied transversely to thelongitudinal axis. In bridge design, beams are the most frequently usedstructural components. The three most common bridge beams are girders,stringers, and floorbeams. Girders are large beams (normally glulam) thatprovide primary superstructure support, most often in beam-type superstructures. Stringers are longitudinal beams that support the bridge deck.

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DESIGN FOR BENDING

They are generally smaller than girders, but there is no clear size definition for either. Floorbeams are transverse beams that directly support thebridge deck or support longitudinal stringers that support the deck. Inaddition to girders, stringers, and floorbeams, other bridge components aredesigned as beams, including components of the deck and railing systems.

Beam design involves the analysis of member strength, stability, andstiffness for four basic criteria: (1) bending (including lateral stability),(2) deflection, (3) horizontal shear, and (4) bearing. Of these four criteria,bending, deflection, and shear can directly control member size, whilebearing will influence the design of supports. Initial beam design is normally based on bending, then checked for deflection and shear. After anappropriate beam size is determined, bearing stresses are checked atsupports to ensure sufficient bearing area.

Beam design requirements discussed in this section are limited to straightor slightly curved (cambered) solid rectangular beams of constant cross-sectional area. Refer to the NDS for design requirements for other beamconfigurations and shapes and for beams with notches or cutouts. Thedesign of beams loaded in combined bending and axial tension or compression is discussed in Section 5.7.

Beam design must consider the strength of the material in bending and thepotential for lateral buckling from induced compressive stress. For positive and negative bending, compression stress occurs in the top and bottomportions of the beam, respectively. Single, simple spans are subjected topositive bending moments only, while multiple continuous spans andcantilevers will be subjected to both positive and negative moments. Thisdistinction is particularly important for stability considerations, and alsowhen the allowable stresses for positive and negative bending are different, as in some combination symbols of glulam beams.

Initial beam design is somewhat of a trial-and-error process. A beam sizeis first estimated, and applied stress is computed and checked against theallowable stress in bending. After a suitable beam is determined fromstrength requirements, it must be verified for lateral stability.

Applied Stress

Applied bending stress in timber beams is determined by the standardformulas of engineering mechanics assuming linear elastic behavior.Stress at extreme fiber in bending, fb is computed by

(5-2)

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Section modulus values for standard sizes of sawn lumber and glulam aregiven in Chapter 16.

Lateral Stability and Beam SlendernessBeams develop compressive stress from induced bending forces. If compression areas are not restrained from lateral movement and rotation, themember may buckle laterally at a bending stress considerably lower than

that normally allowed for the material. The potential for lateral bucklingdepends on the magnitude of applied loads, beam dimensions, and theeffectiveness and frequency of lateral restraint. Lateral stability is mostcritical in long slender beams with a high depth-to-width ratio. It is notcritical in beams where the width of the beam exceeds its depth.

One of the primary factors affecting beam lateral stability is the distancebetween points of lateral support along the beam length. In bridge applications, lateral support is generally provided by cross frames, solid wooddiaphragms, or framing connections that prevent beam rotation and lateraldisplacement (Figure 5-1). The distance between such points of lateralsupport is termed the unsupported length, or When the compression

is zero. For all otherconfigurations,edge is continuously supported along its length,

phragms, or bracing that prevent beam rotation and lateral displacement.is simply the distance between cross frames, dia-

The basis for stability design in beams is the beam slenderness factor Cs,given by

(5-3)

where = effective beam length (in.),

d= beam depth (in.), and

b = beam width (in.).

The effective beam lengthfiguration and loading condition (Figure 5-2). For a single-span beam with

in Equation 5-3 depends on the beam con-

a concentrated load at the center, is computed by

(5-4)

For a single-span beam with a uniformly distributed load, is computed

by

(5-5)

For a single-span beam, or cantilever beam, with any load, is computed

by

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Figure 5-1. - Cross frames fabricated from steel angles are commonly used to providelateral support for large glulam bridge beams.

(5-6)

(5-7)

Equations for computing for other beam configurations and loading

conditions are given in the NDS. For single-span or cantilever beams,Equations 5-6 and 5-7 give slightly conservative results for any loadingcondition and are often used in bridge applications where several concentrated loads are positioned on the span.

Example 5-1 - Beam slenderness factor

A 10-3/4- by 48-inch glulam beam spans 60 feet and supports the threeconcentrated loads shown below. Lateral beam support is provided bytransverse bracing located at the beam ends and at the third points. Com

pute the beam slenderness factor, C .s

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Single-span beam with concentrated load at center

Single-span beam with uniform load

Single-span or cantilever beam with any loading condition

SolutionLateral support is equally spaced along the beam, giving an unsupportedlength of 20 feet. Because the beam is loaded with three concentratedloads, the effective beam length will be computed by Equation 5-6 or

5-7, depending on the ratio of the unsupported length to the beam depth:

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The slenderness factor is computed by Equation 5-3:

This example illustrates a typical case where transverse bracing is equallyspaced and the value ofCapplies to all portions of the beam. In casesswhere the distance varies substantially along the beam length, Cshouldsbe checked for each unsupported length. With few exceptions, however,Cfor the center portion of the beam, where bending stress is highest, willsnormally control.

Allowable StressThe allowable bending stress in beams is controlled either by the size factor CF, which limits bending stress in tension zone, or by lateral stability,which limits bending stress in the compression zone.Adjustments for thesize factor and lateral stability are not cumulative. Therefore, the designermust compute allowable bending stress based on both criteria separately,and the lowest value obtained is used for design. In most bridge beams,allowable bending stress is controlled by CFrather than stability. In addition, beam stability cannot be evaluated until an initial member size isselected. Therefore, it is most convenient and practical to assume that thesize factor controls allowable bending stress and to initially design thebeam based on the allowable stress given by

Values ofCFare normally included in tables of section properties forglulam bending combinations (see Tables 16-3 and 16-4). In addition,most glulam tables include CFas a noted adjustment to the sectionmodulus. This adjusted value, S CF, is included for convenience andxfacilitates design by adjusting for CFduring initial member selection (seeExample 5-3).

After a satisfactory beam size and grade are determined based on theallowable bending stress given by Equation 5-8, the beam must bechecked for lateral stability. Criteria for allowable bending stress related tolateral stability are based on beam slenderness for the following threeranges:

where Ckis a slenderness factor defined later for intermediate beams.

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Short BeamsIn short beams with CS of 10 or less, capacity of the member is controlledby the wood strength in bending rather than by lateral stability. In thiscase, the size factor is the controlling modification factor, and the allowable bending stress computed by Equation 5-8 is used for design.

Intermediate Beams

Intermediate beams have Cgreater than 10, but less than Ckdeterminedsby

In intermediate beams, failure can occur in bending or by torsional buckling from lateral instability. The controlling mode is indicated by thelateral stability of beams factor CL given by

(5-10)

IfCL is less than CF, bending stress is controlled by stability, and CL is thecontrolling modification factor. The allowable bending stress is computed

by

IfCL is greater than CF, bending stress is controlled by strength, and theallowable stress computed by Equation 5-8 is used for design.

Equation 5-9 for lateral stability was developed from theoretical analysesand beam verification tests and is based on the modulus of elasticity of themember. For visually graded sawn lumber, tabulated Evalues are basedon the average modulus of elasticity for the grade and species of materialand represent a coefficient of variation of approximately 0.25. For glulamwith six or more laminations, the coefficient of variation is 0.10 (less thanhalf that for visually graded sawn lumber). To account for this reducedvariability, the NDS allows the designer to use the following modifiedequation for Ck(Equation 5-12), which more accurately reflects the characteristics of glulam:

(5-12)

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modulus of elasticity for this species and grade are obtained fromTable 4A of the NDS:

An initial section modulus based on applied moment and tabulated bend

ing stress is computed as follows:

Rearranging Equation 5-2,

From lumber section properties in Table 16-2, a nominal beam size isselected with a section modulus slightly greater than the required 78.8 in

3.

The closest standard nominal size appears to be 4 inches by 14 inches withthe following properties:

b = 3.5 in.

d= 13.25 in.

S = 102.41 in3

Beam weight = 16.1 lb/ft (based on a unit weight for wood of 50 lb/ft3)

The allowable bending stress is computed using the applicable modifica

tion factors given in Equation 5-8. The size factor, CFis not applicablebecause it only applies to sawn lumber beams that are more than 4 inchesthick. In this case, Equation 5-8 becomes

From Table 5-7, CM= 0.86, and

Next, the applied bending stress is revised to reflect the beam weight of

16.1 lb/ft:

By Equation 5-2,

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fb = 1,207 lb/in2< Fb' = 1,290 lb/in

2, so the initial beam is satisfactory in

bending. The beam must next be checked for lateral stability.

For lateral support at 5-foot intervals,

By Equation 5-5 for a single-span beam with a uniformly distributed load,

By Equation 5-3,

The value C= 12.20 is greater than 10, so further stability calculations aresrequired. From Table 5-7, CMfor modulus of elasticity is 0.97, and

E' = ECM = 1,800,000(0.97) = 1,746,000

By Equation 5-9,

10 < C= 12.20 < Ck= 29.84, so the beam is classified in the intermediatesslenderness range. By Equation 5-10,

The allowable bending stress based on lateral stability is computed byEquation 5-11 using the modification factor CL:

fb = 1,207 lb/in2

< Fb'= 1,277 lb/in2, so the beam size, species, and grade

are satisfactory in bending.

SummaryBased on bending only, the beam will be a nominal 4-inch by 14-inchsurfaced Douglas Fir-Larch beam, visually graded No. 1 in the Joistsand Planks (J&P) size classification. The applied bending stress, fb, is1,207 lb/in

2. The allowable bending stress, Fb', is 1,277 lb/in

2and is con

trolled by lateral stability.

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Example 5-3 - Beam design based on bending; glulam beam

A glulam beam spans 50 feet center-to-center of bearings and supports amoving concentrated load of 20,000 pounds. Determine the required beamsize based on bending for cases where: (A) the beam is laterally supportedat the ends and at the third points, and (B) the beam is laterally supportedat the ends only. The following assumptions apply:

1. Normal load duration under wet-use conditions (glulam moisturecontent will exceed 16-percent in service); adjustments for temperature (Ct) and fire-retardant treatment (CR) are not applicable.

2. The glulam beam is manufactured from visually graded SouthernPine, combination symbol 24F-V2.

Case A: Lateral support is provided at beam ends andat third points

Case B: Lateral support is provided at beam ends only

SolutionThe first step in the design process is to determine the required beam sizebased on bending stress, adjusted by the size factor, CF. The suitability ofthe initial beam size is then checked for each of the two conditions oflateral support.

Tabulated values for bending and modulus of elasticity are obtained from AITC 117-Design. Respective values for the moisture content modification factor are obtained from Table 5-7:

The maximum applied moment is computed with the moving loadpositioned at the span centerline:

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x

An initial beam size is determined using procedures similar to thoseused for sawn lumber beam design. For glulam, however, the size factor,

CF, is included as a noted adjustment to the section modulus (SxCF) inTable 16-4. By Equation 5-8,

Assuming that the applied bending stress equals the allowable bendingstress, Equation 5-2 is rearranged to compute the required value ofS CFdirectly:

Based on the moment from the concentrated load only, an initial value ofS CFis computed:

From Table 16-4, an initial beam size is selected that provides an S CFxvalue slightly greater than 1,563 in

3. It is usually most convenient to

find the closest S CFto that required, then increase the beam depth byx

one or two laminations to account for the beam dead load. In this case, a6-3/4-inch by 41-1/4-inch beam is chosen with the following properties:

S CF= 1,668.9 in3

x

Beam weight = 96.7 lb/ft (based on a unit weight of 50 lb/ft3)

Moment from the beam weight is computed and added to that from theconcentrated load:

M = 250,000 + 30,219 = 280,219 ft-lb

The required S CFvalue is revised:x

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x


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x

C

From Table 16-4, a revised beam size of 6-3/4 inches by 42-5/8 inches isselected with the following properties:

S = 2,044 in3

F= 0.87 Beam weight = 99.9 lb/ft (based on a unit weight of 50 lb/ft

3)

Moment from beam weight is revised and the applied bending stress is computed:

Allowable bending stress is computed by Equation 5-8:

fb = 1,651 lb/in2 < Fb'= 1,670 lb/in2, so the beam is satisfactory in bending,assuming that the size factor controls. The beam is next checked for lateralstability.

Case A: Lateral support at beam ends and at third points

For lateral support at the beam ends and at the third points, the unsupported beam length is equal to one-third the span length:

Because the maximum moment is produced with the moving load atmidspan, the effective beam length is computed using Equation 5-4:

By Equation 5-3,

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The value ofCis greater than 10, so lateral stability must be checkedsfurther. By equation 5-12 for low-variability material,

C= 19.39 < Ck= 25.96, so the beam is in the intermediate beam slender-sness range.

By Equation 5-10,

CL = 0.90 > CF= 0.87, so the size factor reduction is more severe andcontrols the allowable bending stress. The selected beam size is thereforesatisfactory in bending.

Case B: Lateral support at beam ends only

With lateral support at the beam ends only, the unsupported beam lengthequals the span length:

By Equation 5-4,

By Equation 5-3,

The previously computed value Ck = 25.96 is unchanged. In this case,however, Ck= 25.96 < CS = 29.81, so the beam is in the long-beam slenderness range and lateral stability controls design. By low-variabilityEquation 5-14,

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x

fb = 1,651 lb/in2> Fb'= 970 lb/in

2, so the beam must be redesigned. Using

a modified form of Equation 5-2, with the previously computed moment(based on the previous beam size):

From Table 16-4, a revised beam size of 8-1/2 inches by 50-7/8 inches is selected with the following properties: S = 3,666.7 in

3

Beam weight = 150.2 lb/ft (based on a unit weight of 50 lb/ft3)

Moment from beam weight is revised and bending stress is computed:

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DESIGN FOR DEFLECTION

Fb'= 970 lb/in2

This example illustrates the effect of lateral support on beam size requirements. When support along the span is eliminated, the required beam sizeincreases substantially. Additional requirements on the placement anddesign of lateral support for bridge beams are discussed in Chapter 7.

Deflection is the relative deformation that occurs in a beam as it is loaded.Deflection in timber beams results from bending and shear, but sheardeformations are small in comparison to bending deformations and arenormally not considered. Deflection does not seriously affect the strengthof a beam, but it can affect the serviceability and appearance of bridgemembers and the performance of fasteners.

The length of time a load acts on a member influences its long-termdeflection. When loads of relatively short duration are applied, deformation occurs immediately and remains at a relatively constant level for theduration of loading. When the load is removed, the member recoverselastically to the original unloaded position. For permanent loads (deadloads), initial elastic deformation is immediate, but members also developan additional time-dependent, nonrecoverable deformation. This time-dependent deformation, known as creep, develops at a slow but persistentrate and is more pronounced for members seasoned in place or subject tovariations in moisture content and temperature. Creep does not endangerthe safety of the beam, but it can influence the performance, serviceability,and appearance of a structure when it is ignored in design. Thus, the twotypes of deflection considered in timber bridge design are: elastic deflection, and inelastic deflection, or creep.

Deflection EquationsTimber beam deflections are computed by the same engineering methodsused for isotropic, elastic materials. Standard equations based on thesemethods are available in many engineering textbooks and manuals fornumerous beam configurations and loading conditions.

6,27Two of the

most commonly used equations for simple beams are given below inEquations 5-15 and 5-16. Additional equations for more specific bridgeapplications and loads are discussed in Chapters 7, 8, and 9.

For a simply supported beam with one concentrated load at the center ofthe span:

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(5-15)

For a simply supported beam with a uniform load:

(5-16)

where P = magnitude of a single concentrated load (lb),

w = magnitude of uniform load (lb/in),

L = beam span (in.),

E'=ECMCtCR (lb/in2), and

I= moment of inertia about the axis of bending (in4).

Note that the modification factor for duration of load, CD, does not applyto E.

Deflection equations such as 5-15 and 5-16 can be used to accuratelypredict elastic beam deflections. For permanent load deflections,

however, it is necessary to increase computed values to compensate forthe long-term effects of creep. The magnitude of the increase depends onthe type of material and the moisture content of the member at installation.A 50-percent increase in dead load deflection is normally sufficient forglulam and seasoned sawn lumber, while a 100-percent increase is moreappropriate for unseasoned lumber (refer to Appendix F of the NDS foradditional discussions on dead load deflection increases for creep).

Deflection CriteriaAASHTO specifications do not give deflection criteria for timber bridgemembers, and selection of an appropriate deflection limit is a matter of

designer judgment. The acceptable deflection for a member will dependon specific use requirements and may vary among beam types within thesame structure. Deflections in bridge members are important for serviceability, performance, and aesthetics and should not be ignored. From astructural viewpoint, large deflections cause fasteners to loosen and brittlematerials, such as asphalt pavement, to crack and break. In addition,members that sag below a level plane present a poor appearance and cangive the public a perception of structural inadequacy. Deflections from

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moving vehicle loads also produce vertical movement and vibrations thatannoy motorists and alarm pedestrians.

Bridge deflection is normally expressed as a fraction, the denominator ofwhich is obtained by dividing the beam span in inches by the computeddeflection in inches. A deflection ofL/500, for example, indicates adeflection equal to one five-hundredth of the beam span. The larger the

denominator, the smaller the deflection. A brief literature search of bridge-related specifications and publications produced maximum recommendedapplied-load deflection values ranging fromL/200 toL/1,200. For generalbeam design discussed in this chapter, the recommended maximum deflections for timber beams are as follows:

1. For applied (short-term) loads, the maximum deflection shouldnot exceedL/360.

2. For the combination of applied loads and dead load, the maximumdeflection should not exceedL/240, where the portion of the total

deflection from dead load is increased to account for creep.

Additional considerations and recommendations for deflection in timberbridge components are discussed in more detail in Chapters 7, 8, and 9.

CamberCamber is circular or parabolic upward curvature built into a glulam beam,opposite to the direction of deflection. It is intended to offset dead loaddeflection and creep and is introduced during the manufacturing process.It is not feasible to camber sawn lumber beams. The amount of camber forbridge beams depends on the length and number of spans. For single spans

shorter than approximately 50 feet, camber should be a minimum of 1.5 to2.0 times the immediate (elastic) dead load deflection, plus one-half theapplied load deflection.

6For single beam spans equal to or longer than

50 feet and multiple-span beams of any span, camber should be a minimum of 1.5 to 2.0 times the immediate dead load deflection (multiple-spanbridge beams are normally cambered for dead loads only to obtain acceptable riding qualities for vehicle traffic).

Camber is specified by the designer as a vertical centerline offset to thehorizontal line between points of bearing (Figure 5-3). The glulam manufacturer will determine an appropriate radius of curvature based on offsetdistances and fabrication limitations. On multiple-span continuous beams,camber may vary along the beam and should be specified for each spansegment. More specific information on cambering practices and limitations can be obtained from glulam manufacturers and the AITC.

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Figure 5-3.- Camber for glulam beams is specified as an upward vertical offset at the spancenterline.

Example 5-4- Beam deflection and camber

For the glulam beam of Example 5-3, Case A, determine the deflectionfrom the 20,000-pound moving load and the camber required to offset

deflection from the beam weight. The beam spans 50 feet, measures6-3/4 inches by 42-5/8 inches, and is manufactured from visually gradedSouthern Pine, combination symbol 24F-V2.

Solution:The tabulated modulus of elasticity for a 24F-V2 Southern Pine beam isobtained fromAITC 117-Design:

E= 1,700,000 lb/in2

x

The allowable modulus of elasticity is computed using the applicable CMvalue from Table 5-7:

E' = E CM= 1,700,000(0.833) = 1,416,100 lb/in2

x

From Table 16-4 for a 6-3/4-inch by 42-5/8-inch Southern Pine beam:

I= 43,562.8 in4

x

Beam weight = 99.9 lb/ft (based on a beam weight of 50 lb/ft3)

Deflection for the 20,000-pound moving load is computed with the load atmidspan by Equation 5-15:

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DESIGN FOR SHEAR

Expressing the deflection as a ratio of the bridge span,

For the beam weight of 99.9 lb/ft, deflection is computed byEquation 5-16:

Camber of approximately 1/2-inch will be specified at centerline, which isapproximately twice the beam dead load deflection.

Beams develop internal shear forces that act perpendicular and parallel tothe longitudinal beam axis. In timber beams, horizontal shear rather than

vertical shear will always control design. As discussed in Chapter 3,horizontal shear forces produce a tendency for the upper portion of thebeam to slide in relation to the lower portion of the beam, with shearstresses acting parallel to the grain of the member. The maximum intensityof horizontal shear in rectangular beams occurs at the neutral axis and isproportional to the vertical shear force, V. In bridge applications, horizontal shear generally controls beam design only on relatively short, heavilyloaded spans.

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Shear requirements in AASHTO and the NDS apply at or near the supports for solid beams constructed of such materials as sawn lumber,glulam, or mechanically laminated lumber. Shear design for built-upcomponents containing load-bearing connections at or near supports, suchas between a web and chord, must be based on tests or other techniques.

Applied Stress

The applied stress in horizontal shear depends on the magnitude of the vertical shear and the area of the beam. Applied stress in square or rectan gular timber beams is computed by Equation 5-17:

(5-17) where f= unit stress in horizontal shear (lb/in

2),v

V= vertical shear force (lb),

b = beam width at the neutral axis (in.),

d= beam depth (in.), and

A = beam cross-sectional area (in2).

Equation 5-17 does not apply (1) at notches or joints, (2) in regions wherethe beam is supported by fasteners, or (3) when hanging loads are locatedat or near the supports. For these conditions, refer to AASHTO and theNDS.

The magnitude off, given by Equation 5-17 is based on the value of the

vertical shear force, V. Unlike the situation in other construction materials,where the maximum vertical shear is computed at the face of the supports,in timber beams the maximum intensity of horizontal shear is produced bythe maximum vertical shear force occurring at some distance from thesupport. This distance depends on the type of applied loading; differentdistances are used for moving loads and for stationary loads.

Current AASHTO requirements (AASHTO 13.3.1) specify that horizontalshear in beams from moving (vehicle) loads be computed from the maximum vertical shear (V) occurring at a distance from the support equal tothree times the beam depth (3d, or the span quarter point (L/4), whichever

is less (Figure 5-4). The moving loads are positioned on the beam toproduce the maximum vertical shear at this location (Chapter 6). Forstationary loads (such as dead load), vertical shear is computed at a distance from the support equal to the beam depth, d, and all loads occurringwithin the distance dfrom the supports are neglected. For sawn lumber,shear design requirements given in the NDS vary somewhat based on thebeam configuration, loading condition, and wood species. Refer to thelatest edition of the NDS for additional shear criteria for sawn lumber.

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increase is commonly used for mechanically laminated lumber and dimension lumber with loads applied perpendicular to the wide face. Additionalinformation on application of the shear stress modification factor is discussed in Chapters 7 and 8.

Table 5-12.- Shear stress modification factor for sawn lumber.

ExampIe 5-5- Horizontal shear in a sawn lumber beam

Determine the adequacy of the beam in Example 5-2 for horizontal shear.The beam measures 4 inches by 14 inches and is surfaced Douglas Fir-

Larch, visually graded No. 1 in the J&P size classification. It spans 15 feetand supports a uniform load of 350 lb/ft.

SolutionTabulated horizontal shear stress for No.1 Douglas Fir-Larch is obtainedfrom Table 4A of the NDS (note that the tabulated shear stress for lumber2 to 4 inches thick is the same for all grades):

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F= 95 lb/in2

v

Allowable shear stress is computed by Equation 5-18 using the CMvalueobtained from Table 5-7,

The allowable stress in horizontal shear could be increased by the shearstress modification factor (Table 5-12) if the beam were free of shake,splits or checks, or if the length of such characteristics was known. Forlumber bridge beams of this type, it is common for some beam checking tooccur, however, its magnitude cannot be accurately predicted. Therefore,no adjustment by the shear stress modification factor will be used.

From Example 5-2, the beam weighs 16.1 lb/ft and has actual dimensionsof 3.5 inches by 13.25 inches. The total load acting on the beam is equal tothe 350 lb/ft applied load plus the beam weight of 16.1 lb/ft, for a total of366.1 lb/ft. For a uniformly distributed load, the maximum vertical shear

force, V, is computed at a distance from the support equal to the beamdepth, d, and all loads acting within a distance dfrom the supports areneglected:

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Horizontal shear stress is computed by Equation 5-17:

A = (3.5 in.)(13.25 in.) = 46.38 in2

f= 76 lb/in2 < F ' = 92 lb/in2, so horizontal shear is acceptablev v

Example 5-6- Horizontal shear in a glulam beam.

Check the adequacy of the glulam beam in Example 5-3, Case A, for horizontal shear. The beam measures 6-3/4 inches by 42-5/8 inches and ismanufactured from visually graded Southern Pine, combination symbol24F-V2. It spans 50 feet and supports a moving concentrated load of20,000 pounds.

The tabulated stress for horizontal shear for a 24F-V2 beam is obtainedfrom AITC 117--Design,

CAllowable shear stress is computed by Equation 5-18 using the applicable

Mvalue obtained from Table 5-7:

CM= 0.875

F ' = F CM = 200(0.875) = 175 lb/in2

v vx

In this case the beam supports two loads; the uniform load from the beamweight and the moving concentrated load. Maximum vertical shear fromthe uniformly distributed beam weight is computed at a distance from the

support equal to the beam depth, d, and all loads acting within a distance dfrom the supports are neglected. For the moving concentrated load, maximum vertical shear is computed at a distance from the support equal tothree times the beam depth, 3d, or the span quarter point,L/4, whicheveris less.

For the uniformly distributed beam weight of 99.9 lb/ft and a beam depthof 42.63 inches,

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DESIGN FOR BEARING

For the moving concentrated load of 20,000 lb,

3d


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Applied StressApplied bearing stress is computed by

(5-19)

where = unit stress in compression perpendicular to grain (lb/in2),

R = reaction or bearing force at the support (lb), and

A = net bearing area (in2).

When computingfact that as the beam bends the pressure on the inner edge of the bearing is

at the end of a beam, no allowance is made for the

greater than that at the end of the beam.

Allowable StressThe allowable stress for bearing perpendicular to grain is equal to thetabulated stress adjusted by all applicable modification factors, except

the duration of load factor, CD, as computed by

When beam bearing is not perpendicular to grain (Figure 5-5), allowablestress must be computed for compression at an angle to the grain using theHankinson Formula (Equation 5-21):

(5-21)

where F ' = allowable stress in compression at an angle to thengrain (lb/in

2),

Figure 5-5. -- Beam bearing at an angle to the grain.

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= angle between the direction of load and the direction ofgrain (degrees).

Values of given in the NDS and AITC 117-Design apply to bearingsof any length at beam ends and to all bearings 6 inches or more in lengthat other locations. Refer to the NDS for required adjustments in tabulatedstress for bearings less than 6 inches long at locations between beam ends.

Example 5-7 - Beam bearing

For the glulam beam of Example 5-3, Case A, determine the requiredbearing length and the bearing stress in compression perpendicular tograin. The beam spans 50 feet center-to-center of bearings, is 6-3/4 incheswide and supports a moving concentrated load of 20,000 pounds. It ismanufactured from visually graded Southern Pine, combination symbol24F-V2.

SolutionThe tabulated stress in compression perpendicular to grain for a 24F-V2Southern Pine beam is obtained from AITC 117-Design:

The allowable compression perpendicular to grain is computed usingEquation 5-20 and the applicable CMvalue from Table 5-7:

The maximum reaction at the beam bearing is equal to the sum of thereactions from the moving concentrated load and the beam weight. Themaximum reaction from the moving concentrated load occurs when theload is placed over one support:

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The reaction from the beam weight is the same at both supports:

Rearranging Equation 5-19, the minimum required bearing area is computed for the maximum reaction by substituting

For a beam width of 6-3/4 inches, the required bearing length is computed

by dividing the bearing area by the bearing width:

A bearing length of 10 inches is selected and applied stress is computed byEquation 5-19:

= 345 lb/in2, so the bearing is satisfactory. For a

center-to-center span of 50 feet, a beam length of 50 feet 10 inches will berequired.

5.5 DESIGN OF TENSION MEMBERS

A tension member is a structural component loaded primarily in axialtension. In bridge design, tension members are used mostly as truss elements and occasionally as bracing (Figure 5-6). The direction of loading intension members should always be parallel to the grain of the member.

Timber is weak in tension perpendicular to the grain, and loading conditions that produce stress in this direction should be avoided. Whenloading conditions that induce tension perpendicular to the grain do exist,mechanical reinforcement must be designed to carry the load.

Discussions in this section apply to members loaded in axial tension only.Design criteria for members loaded in combined axial tension and bendingare given in Section 5.7.

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Figure 5-6.- Tension members in bridge applications are most common in trusses. Thistimber truss, located at Sioux Narrows, Ontario, Canada, spans 210 feet and is reputed tobe the longest clear-span timber bridge in the world.

APPLIED STRESS Applied stress in tension is computed by Equation 5-22:(5-22)

where P = axial load applied to the member (lb), and

A = net cross-sectional area of the member (in2).

The net area,A, in Equation 5-22 is the gross area of the member minusthe projected area of fastener holes or cuts that reduce the section. Requirements for determining net area for various fasteners are discussed inSection 5.8.

ALLOWABLE STRESS Allowable stress in tension equals the tabulated stress for tension parallelto grain, Ft, adjusted by all applicable modification factors. This is computed by

(5-23)

For sawn lumber, values ofFtfor members 2 to 4 inches thick, and5 inches and wider, apply to 5- and 6-inch widths only. When wider members are used, a reduction in tabulated stress ranging from 0.9 to 0.6 is

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required by footnotes to the NDS Table 4A. When glulam is used, themost economical tension members are generally selected from the axialcombinations given inAITC 117-Design.

Example 5-8- Glulam tension member

A glulam truss member carries an axial tension load of 25,000 pounds.

The ends of the member are attached to steel plates with a single row of1-inch-diameter bolts aligned in the longitudinal direction. Design thistruss member, assuming the following:

1. Normal load duration under wet-use conditions; adjustments fortemperature (Ct) and fire-retardant treatment (CR) are notapplicable.

2. Bolt holes at member ends are 1/16 inch larger than the boltdiameter.

3. Glulam is manufactured from visually graded western species.

SolutionThe design of a tension member starts with either the selection of a glulamcombination symbol or a standard member width. In this example, combi

nation symbol No. 2 is selected and design will involve determining therequired member size.

The tabulated stress for tension parallel to grain is obtained for combination symbol No. 2 from AITC 117-Design:

Ft= 1,250 lb/in2

The allowable stress for tension parallel to grain is computed byEquation 5-23 using the CMvalue obtained from Table 5-7:

Next, Equation 5-22 is rearranged to compute an initial member areabased on the applied load and the allowable stress in tension parallel tograin:

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The required member depth is obtained for several standard glulam widthsby dividing the required area by the standard width, then rounding thedepth up to the next standard depth (based on a 1-1/2-inch laminationthickness for western species). For three standard glulam widths:

Initial selection of a member width and depth is a matter of designerjudgement and depends on size and economic considerations. In this case,the 5-1/8-inch width is selected and the gross member area is computed:

The net area used for design is equal to the gross area minus the projected area of bolt holes. Assuming that bolts pass through the narrow(5-1/8-inch) dimension,

By Equation 5-22,

ft=987 in2< Ft' = 1,000 lb/in

2, so a 5-1/8-inch wide by 6-inch deep

combination symbol No. 2 member is satisfactory.

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5.6 COLUMN DESIGNA column is a structural component loaded primarily in axial compressionparallel to its length. In bridge design, columns are used as supportingcomponents of the substructure, truss elements, and bracing (Figure 5-7).The three general types of columns are simple solid columns, spacedcolumns, and built-up columns (Figure 5-8). Simple solid columns consist

of a piece of sawn lumber or glulam. Spaced columns consist of two ormore parallel pieces that are separated and fastened at the ends and at oneor more interior points by blocking. Built-up columns consist of a numberof solid members joined together with mechanical fasteners. The mostcommon columns for timber bridges are simple solid columns constructedof sawn lumber, glulam (axial combinations), timber piles, or poles.Although spaced and built-up columns may be used for truss elements orother components, they are not common in modem bridge applications.

The column design requirements in this section are limited to simple solidcolumns of constant cross-sectional area. Loads are applied concentrically,

and design is based on the stresses and instability from axial compressionand end-grain bearing stress at column ends. Columns loaded in combinedcompression and bending are discussed in Section 5.7 of this chapter. Foradditional information on built-up, spaced, and tapered solid columns,refer to theNDS and theAITC Timber Construction Manual.

Figure 5-7.- Timber columns are common in bridge substructures such as these bents(photo courtesy Wheeler Consolidated, Inc.).

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DESIGN FORCOMPRESSION

Simple solid Simple solid column of column of

sawn lumber glulam Spaced Built-up

column of column ofnailed lumber bolted lumber

Figure 5-8. - General classes of timber columns.

Compression in timber columns can induce failure by crushing the woodfibers or by lateral buckling (deformation). The first step in column designis to estimate an initial member size and compute applied stress (severaliterations may be required to arrive at a suitable section). After an initialcolumn size is selected, the column slenderness ratio is computed, whichserves as the basis for design in compression. From the slenderness ratio,allowable stress is determined from equations given in the NDS andchecked against the applied stress.

Applied StressApplied column stress in compression parallel to grain, fc, is computed by

(5-24)

where P = the total compressive load supported by the column (lb), and

A = the cross-sectional area of the column (in2).

The value ofA used in Equation 5-24 depends on the location of fastener

holes that reduce the column section. When the reduced section occurs atpoints of lateral support, failure occurs by wood crushing, and the grosscolumn area is used without deductions for fastener holes. At locationsaway from points of lateral support, failure may occur by column buckling, and the net column area (gross column area minus fastener holes) isused. Refer to Section 5.8 for details on computing net area for differentfastener types.

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Column Slenderness RatioThe slenderness ratio of a column provides a measure of the tendency ofthe column to fail by buckling from insufficient stiffness, rather than bycrushing from insufficient strength. It is expressed as the ratio of theunsupported column length to its least radius of gyration and is computedfor timber in the same manner as for other materials. For convenience indesign, however, the slenderness ratio for square or rectangular simple

solid columns is given in terms of the column cross-sectional dimension,rather than the radius of gyration, and is computed by

Slenderness ratio = (5-25)

where = effective column length (in.), and

d= cross-sectional dimension corresponding to (in.).

The effective column length in Equation 5-25 is the distance between two points along the column length at which the member is assumed to buckle in the shape of a sine wave. It is computed as the product of the unsupported column length and the effective buckling length factor given by

(5-26) where K= effective buckling length factor, ande

= unbraced length between points of lateral support along the column length.

Values ofKare given in Table 5-13 for various conditions of end fixityeand lateral translation at column ends or intermediate points of lateralsupport. In most applications, timber columns with square-cut ends arefixed against translation but not rotation (approximately pinned connections), and the value ofKis 1.0. Conditions may be encountered in designewhere restraint is more or less than this condition, and Ke must be adjustedaccordingly based on designer judgment. Additional discussion on effective buckling length factors is given in Appendix N of the NDS.

The slenderness ratio provides an indication of the mode of failure and isthe basis for determining the allowable design stress. If a column is loadedto failure by buckling, the buckling will always occur about the axis with

the largest slenderness ratio. The task of the designer is to determine thecontrolling slenderness ratio for a given column configuration. For arectangular column with the same unbraced length in both directions, thecritical slenderness ratio can be determined by inspection (Figure 5-9 A).In this case, the column will obviously buckle about the weaker (y) axis,and that is the only slenderness ratio that must be computed (for bucklingabout they axis the column deflects in thex direction). For column configurations where the unbraced length is not the same in both directions,

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Table 5-13. - Effective buckling length factor, eK .

the critical slenderness ratio cannot be determined by inspection andthe designer must compute slenderness ratios for both directions(Figure 5-9 B). Depending on the spacing of lateral support, conditionsmay exist where the column design is controlled by buckling about thestrong axis.

Allowable StressThe allowable compressive stress for square or rectangular simple solidcolumns is computed from equations given in the NDS. These equations

are based on the column slenderness for three ranges:

where Kis a slenderness factor defined later in this section for intermediate columns.

The NDS equations have been modified to incorporate the use of thecolumn dimension (d) rather than the radius of gyration (r). They may be

is used in place ofused for nonrectangular cross sections by substituting 3.46rfor

special case of a round column, the NDS states that the load on a roundwhen determining the column-length class). For the

cross-sectional area. For round columns, the dused in determing the ratio is 0.866 times the diameter of the round column.

5-52

column may be taken as the same as that for a square column of the same


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Short ColumnsShort columns are columns with a slenderness ratio of 11 or less. In shortcolumns, the capacity of the member is controlled by the strength incompression parallel to grain, and failure always occurs by crushing of thewood fibers. Allowable stresses for short columns are equal to the tabulated stress in compression parallel to grain adjusted by applicable modification factors, as given by

(5-27)

Intermediate ColumnsIntermediate columns have a slenderness ratio greater than 11 but less thanKas determined by

(5-28)

In intermediate columns, failure can occur by crushing of the wood fibersor by lateral buckling, or both. The allowable stress for intermediatecolumns is the tabulated stress in compression parallel to grain adjusted byapplicable modification factors, including the lateral stability of columnsfactor, CP, and is computed by

(5-29)

where

(5-30)

In addition to Equation 5-29, the NDS gives optional column designadjustments for low variability materials (such as glulam) that are similarto those previously discussed for beams. For additional information onthese equations, refer to Appendix G of the NDS and the AITC TimberConstruction Manual.

Long ColumnsLong columns are columns with a slenderness ratio greater than Kand lessthan or equal to 50 (the maximum slenderness ratio allowed by the NDSfor any column is 50). In long columns, the strength of the member is controlled by stiffness, and failure occurs by lateral buckling. The allowabledesign stress for long columns is given by

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(5-3 1)

DESIGN FOR BEARING Column design must also consider bearing on the end grain of the member, given by

(5-32) where f= end-grain bearing stress from applied loads (lb/in

2),g

P = total applied load (lb), and

A = net area in bearing (in2).

The tabulated stress for end grain in bearing is specified in Table 2B of theNDS for sawn lumber and in Tables A-1 and A-2 of AITC 117-Design

for glulam. The tabulated stress for sawn lumber is given for wet-serviceand dry-service conditions. For glulam, tabulated stress is for dry-serviceconditions and must be modified when the moisture content of the member is expected to exceed 16 percent in service (as in most bridge applications). Tabulated end-grain bearing stress is computed for sawn lumberand glulam as follows:

For sawn lumber,

(5-33)

For glulam,

(5-34)

When the bearing stress computed by Equation 5-32 exceeds 75 percent ofthe allowable stress computed by Equations 5-33 or 5-34, the NDS requires that the bearing be on a metal plate or strap, or on other durable,rigid, homogeneous material of adequate strength.

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Example 5-9. - Column design; sawn lumber

A square, sawn lumber column is 6 feet high and supports a concentricload of 35,000 pounds. Lateral support for the column is provided bypinned connections at the column ends only. Design this column, assuming the following:

1. Normal load duration and wet- use conditions; adjustments for temperature (Ct) and fire- retardant treatment (CR) are not required.

2. The column is S4S Douglas Fir-Larch, visually graded No. 1 to WCLIB rules in the Posts and Timbers (P&T) size classification.

SolutionThe first step in column design is todetermine an initial column size. Sincecolumn dimensions are initially unknown, it is usually assumed that thecolumn is in the short column slenderness range, and the allowable stress in compression parallel to grain iscomputed using Equation 5-27:

From the NDS Table 4A for No. 1 Douglas Fir-Larch in the P&T sizeclassification,

F= 1,000 lb/in2

c

C

From Table 5-7,

M= 0.91

Substituting values,

An initial column area is obtained by dividing the applied load by F ':c

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From Table 16-2, the smallest square lumber size that meets the minimumarea requirement is 8 inches by 8 inches, with the following properties:

b = 7.5 in.

d = 7.5 in.

A = 56.25 in2

The column slenderness ratio must next be computed to determine theactual column slenderness range. The effective column length is computedby Equation 5-26 using an unbraced length of 6 feet and an effectivebuckling length factor, Ke, of 1.0 for the pinned ends:

The column slenderness ratio is computed by Equation 5-25:

so the column is in the short column slenderness rangeas initially assumed. Applied stress is computed by Equation 5-24:

f= 622 lb/in2< F '= 910 lb/in

2, so the column size is satisfactory.c c

Although normally not a controlling factor in column design, end grain inbearing stress should also be checked. From NDS Table 2B for wet-useDouglas Fir-Larch,

F= 1,340 lb/in2

g

By Equation 5-33,

F' = F CD = 1,340( 1.0) = 1,340 lb/in2

g g

0.75F '= 0.75 (1,340) = 1,005 lb/in2

g

Assuming a unit weight for wood of 50 lb/ft 3

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By Equation 5-32,

f= 624 lb/in2< 0.75F ' = 1,005 lb/in

2, so end-grain bearing is satisfactory,

and bearing on a steel plate or other rigid, homogeneous material is notrequired.

g g

SummaryThe column will be nominal 8-inch by 8-inch surfaced Douglas Fir-Larch,visually graded No. 1 in the P&T size classification. The column isclassified in the short column slenderness range andf= 622 lb/in

2< F ' =c c

910 lb/in2. End-grain bearing stress is less than 75 percent of the allowable

value, so special steel bearing plates are not required.

Example 5-10- Glulam column designA glulam column is 17 feet long, 8-1/2 inches wide and 12-3/8 inchesdeep. Determine the column capacity for concentric loading when (A) thecolumn is laterally supported at the ends only, and (B) the column islaterally supported at the ends and at midheight along the 12-3/8-inchdimension. The following assumptions apply:

1. Normal load duration under wet-use conditions; adjustments fortemperature (Ct) and fire-retardant treatment (CR) are notapplicable.

2. Glulam is visually graded Southern Pine, combination symbolNo. 47.

3. All support connections are pinned.4. End-grain bearing is on a steel plate.

SolutionThe procedure for determining the allowable load for each support condition will first involve computing the column slenderness range. From this,the allowable unit stress and load will be determined.

Tabulated values for compression parallel to grain and modulus of elasticity are obtained fromAITC 117--Design. Respective values for the moisture content modification factor are obtained from Table 5-7:

F= 1,900 lb/in2

CM= 0.73

E= 1,400,000 lb/in2

CM= 0.833

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The allowable load is the product of the column area and F ':c

Case B: Lateral support at column ends and at midheight along the12-3/8-inch dimension

With lateral support at the column ends and at midheight along one axis,the slenderness ratio must be checked for both axes. About thex-x axis:

K= 1.0e

d=12.38 in.

About they-y axis:

K= 1.0e

d= 8.5 in.

x-x axis) will controlThe largest slenderness ratio of 16.48 (about thedesign. By previous calculations K = 19.46 > =16.48, so the columnis in the intermediate range.

The lateral stability of columns factor, Cp, is computed by Equation 5-30;

Allowable stress in compression parallel to grain is computed byEquation 5-29:

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The allowable load is the product of the column area and F ':c

P =A(Fc') = 105.2(1,151) = 121,085 lb

Check End-Grain Bearing

The tabulated stress for end grain in bearing is obtained from AITC 117Design:

F= 2,300 lb/in2

g

The allowable stress is computed using Equation 5-34:

F '= 1,311 lb/in2is greater than previously computed values of F ', so

g c

bearing stress will not control.

SummaryThe allowable compression parallel to grain and maximum load for bothcolumn support cases are as follows:

Case A: Column laterally supported at ends only

F '= 607 lb/in2

c

Maximum allowable load = 63,856 lb

Case B: Column laterally supported at ends and at midheight along

the 12-3/8-inch dimension

F '= 1,151 lb/in2

c

Maximum allowable load = 121,085 lb

This example illustrates the effect that lateral support can have on allowable column loading. When additional support is added at midheight,along the 12-3/8-inch dimension, the allowable load nearly doubles.

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5.7 DESIGN FOR COMBINED AXIAL AND BENDING FORCESOne or more loads acting on a column, beam, or other structural membermay induce a combination of axial and bending stresses that occur simultaneously. In bridge design, combined loading most commonly occurs asaxial compression and bending acting on supporting columns of thesubstructure (Figure 5-10). Even in columns designed for concentric loads,

small eccentricities are created because of construction tolerances, slightmember curvature, and material variations. Bending stress also occurswhen columns are subjected to transverse loads from wind or earthquakes(see Chapter 6). Other conditions involving combined compression andbending or combined tension and bending are less common in bridgeapplications, but may occur in truss members or other components.

The design requirements discussed in this section are for combined axialtension or compression acting simultaneously with bending. It is assumedthat bending occurs about one axis and that all loads are applied directly tothe member. For cases involving axial loads with biaxial bending or loads

acting through brackets attached to the member side, refer to referenceslisted at the end of this chapter.

6,7,8,21,26,34

When members are subjected to simultaneous axial and bending loads, theresulting stress distribution is approximately the sum of the effects of theindividual loads. In combined tension and bending, the effect is to reducethe compressive stress on one side of the member and increase the tensilestress on the other side. For combined compression and bending, tensilestress is reduced on one side and compressive stress is increased on theother. The case of combined compression and bending is critical because

the higher compression increases the potential for lateral buckling of themember.

Combined stresses are evaluated using an interaction formula. In generalterms, the interaction formula contains two expressions, one for the capacity in axial loading and one for the capacity in bending. In its basic form,the interaction formula is expressed by

(5-35)

Each of the expressions in Equation 5-35 can be thought of as representingthe portion of the total member capacity taken by the respective axial orbending stress. The axial portion of the formula is the ratio of the appliedaxial stress to the allowable axial stress, assuming the member is loaded

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Figure 5-10.- Members subjected to combined axial and bending forces are most commonin bridge substructures. The vertical posts of this abutment support compressive loadsfrom the superstructure and lateral loads from the earth pressure on the abutment wall.

with axial forces only. The bending portion is the ratio of the appliedbending stress to the allowable bending stress, assuming the member isloaded with bending forces only. The sum of these expressions cannotexceed 1.0, or 100 percent of the member capacity.

When selecting a glulam member for combined axial and bending stresses,

the designer should consider the relative magnitude of each type of stress.If tension or compression is the predominant stress, axial combinations areusually most economical. When bending is the predominant stress, bending combinations may be more appropriate.

COMBINED BENDING AND When members are loaded in combined axial tension and bending, theAXIAL TENSION interaction equations that must be satisfied for design are given by

(5-36)

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COMBINED BENDING ANDAXIAL COMPRESSION

In applying the interaction formulas, tension stress is computed for atension member, as discussed in Section 5.5, and bending stress is com

puted for a beam, as discussed in Section 5.4. Considerations for tensionare relatively straightforward; however, for bending, the member must bechecked for strength in the tension zone and stability in the compressionzone. In beam design, the size factor, CF, applies to the tension side of themember where stresses from combined loading are greater than those frombending alone. As a result, CFis always used as a modification factor inEquation 5-36. The lateral stability of beams factor, CL, affects the compression side in bending where stresses from combined loading are reduced by the axial tension. When conditions of lateral support are suchthat the member is classified as an intermediate or long beam, and CLrather than CFcontrols beam design, the member must also meet the

stability requirements given in Equation 5-37.

Members subjected to combined axial compression and bending arecommon in bridge design and are frequently referred to as beam columns.This type of loading is more critical than combined tension and bendingbecause of the potential for lateral buckling and the additional bendingstress created by the P-delta effect. The P-delta effect is produced whenbending loads cause the axially loaded member to deflect along its longitudinal axis. When this occurs, an additional moment is generated by theaxial load, P, acting over a lever arm equal to the deflected distance

(Figure 5-11). The potential magnitude of the P-delta moment depends onthe stiffness of the member and is not computed directly; however, theinteraction equations for combined compression and bending includeadditional terms to compensate for this effect.

The exact analysis of a member with combined axial compression andbending can be

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